Micro-electromechanical (MEMs) resonators can provide small form factor, ease of integration with conventional semiconductor fabrication techniques and high f·Q products. High frequency and high-Q width-extensional mode silicon bulk acoustic resonators (SiBARs) and film bulk acoustic wave resonators (FBARs) have demonstrated atmospheric Q factors in excess of 10,000 at or above 100 MHz, with moderate motional resistances. Such resonators are disclosed in an article by S. Pourkamali et al., entitled “Low-Impedance VHF and UHF Capacitive Silicon Bulk Acoustic Wave Resonators—Part I: Concept and Fabrication,” IEEE Trans. On Electron Devices, Vol. 54, No. 8, pp. 2017-2023, August (2007), the disclosure of which is hereby incorporated herein by reference.
The resonant frequency of silicon micro-electromechanical resonators is dependent on the physical dimensions of the resonating structure. This causes the resonant frequency of those resonators to deviate from a designed target value in response to variations in photolithography, etching and film thickness. For example, as described in an article by G. Casinovi et al., entitled “Analytical Modeling and Numerical Simulation of Capacitive Silicon Bulk Acoustic Resonators,” IEEE Intl. Conf. on Micromechanical Systems (2009), a 2 μm variation in thickness of a 100 MHz width-extensional mode SiBAR can cause a 0.5% variation in its center frequency, while lithographic variations of ±0.1 μm in the width of the resonator can cause an additional 0.5% variation in frequency.
Unfortunately, even when efforts to reduce the adverse effects of variations in photolithography, etching and film thickness on resonant frequency are successful, additional changes in resonant frequency may occur in response to changes in operating temperature. These temperature-based changes in resonant frequency can be reduced using modified fabrication processes and active compensation circuits. However, because circuit-based compensation techniques typically increase the complexity and power requirements of resonator devices, passive fabrication-based compensation techniques based on the intrinsic properties of the resonator materials are generally preferable to circuit-based compensation techniques. Conventional passive compensation techniques are disclosed in U.S. Patent Publication Nos. 2010/0032789 to Shoen et al., entitled “Passive Temperature Compensation of Silicon MEMS Devices;” and 2009/0160581 to Hagelin et al., entitled “Temperature Stable MEMS Resonator.”